$\DeclareMathOperator{\span}{span}$ $\DeclareMathOperator{\co}{H}$ $\newcommand{\kk}{\mathbb{F}}$ $\newcommand{\qq}{\mathbb{Q}}$ $\newcommand{\zz}{\mathbb{Z}}$ $\newcommand{\rr}{\mathbb{R}}$ $\newcommand{\semi}{\hat{\chi}_2}$ $\newcommand{\ori}[1]{\textbf{(O$_{\pmb{#1}}$)}}$ $\newcommand{\nori}[1]{\textbf{(NO$_{\pmb{#1}}$)}}$ $\newcommand{\rarr}{\rightarrow}$Let $M$ be a smooth connected $d$-manifold (without boundary), and write $\span(M)$ for the maximum number of linearly independent vector fields on $M$. By Poincaré-Hopf, we know that if $M$ is closed, then $\span(M) \geq 1$ if and only if $\chi(M) = 0$. It is also known that every open $M$ satisfies $\span(M) \geq 1$. I am interested in the characterization of the condition $\span(M) \geq 2$. This has been achieved for closed manifolds with works of several people (over the years ~1965-90), as I will explain.
Notation for closed $M$: Given a field $\kk$, write $b_j(M;\kk) := \dim_\kk{\co_j(M;\kk)}$. Write $$\semi(M;\kk) := \left( \sum_{j \geq 0} b_{2j}(M;\kk) \!\!\!\mod{\!2}\right) \in \kk_2$$ for the Kervaire semi-characteristic over $\kk$ (it is always a mod-2 number). For each $0 \leq j \leq d$, write $w_j(M) \in \co^j(M;\kk_2)$ for the $j$-th Stiefel-Whitney class. Write $[M] \in \co_d(M;\kk_2)$ for the $\kk_2$-fundamental class and $$\langle-,[M] \rangle \colon \co^k(M;\kk_2) \rarr \co_{d-k}(M;\kk_2)$$ the associated PD isomorphism. If $M$ is oriented and $d \equiv 0 \!\pmod{4}$, let $\sigma(M)$ denote its signature.
For the characterization of closed $M$ with $\span(M) \geq 2$, first note that when $d=2$ we only have the $2$-torus. Assuming $d \geq 3$, then $M$ has $\span(M) \geq 2$ if and only if in addition to $\chi(M) = 0$, it satisfies one of the following conditions:
$\ori{0}$ : $M$ is orientable, $d \equiv 0\!\pmod{4}$, and $\sigma(M) \equiv 0\!\pmod{4}$.
$\ori{1}$ : $M$ is orientable, $d \equiv 1\!\pmod{4}$, $w_{d-1}(M) = 0$, and $\semi(M;\rr) = 0$
$\ori{2,3}$ : $M$ is orientable and $d \equiv 2,3 \!\pmod{4}$.
$\nori{0,2}$ : $M$ is non-orientable, $d$ is even, and writing $\zz_{w_1(M)}$ for the orientation sheaf, the twisted Bockstein $$\beta^{*}\colon \co^{d-2}(M;\kk_2) \rightarrow \co^{d-1}(M;\zz_{w_1(M)})$$ sends $w_{d-2}(M)$ to $0$.
$\nori{1}$ : $M$ is non-orientable, $d \equiv 1\!\pmod{4}$, $w_1(M)^2 = 0 = w_{d-1}(M)$, and $$\semi(M;\kk_2) = \langle w_2(M)w_{d-2}(M), [M] \rangle \in \kk_2 \, .$$
$\nori{3}$ : $M$ is non-orientable, $d \equiv 3\!\pmod{4}$, and $w_1(M)^2 = 0$.
$\nori{1,3}$ : $M$ is non-orientable, $d$ is odd, $w_1(M)^2 \neq 0$, and $w_{d-1}(M) = 0$.
[I can give precise references with a bit more work, but for now I will note that $\ori{0}$ is due to Frank and independently Atiyah for $d > 4$ and due to Randall when $d=4$, $\ori{1}$ is due to Atiyah, $\ori{2,3}$ is due to E. Thomas, $\nori{0,2}$ is due to Pollina, $\nori{1}$ and $\nori{3}$ are due to Randall, $\nori{1,3}$ is due to Mello.]
For open manifolds, I could only find the following: Non-orientable surfaces necessarily have $\span = 1$, and open orientable manifolds of dimension 2,3 are parallelizable.
Can we similarly characterize open $d$-manifolds $M$ with $\span(M) \geq 2$? For instance, can the method in this answer be adapted to show that $\span(M) \geq 2$ whenever $M$ is orientable and $d \equiv 3\!\pmod{4}$?
